42 SIA diffusion kinetics With all the derived mechanisms and associated

42 sia diffusion kinetics with all the derived

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4.2. SIA diffusion kinetics With all the derived mechanisms and associated barriers in section  3 , we employ the kMC simulation to study the SIA diffusion in hcp Zr. According to transition state theory, the transition rate can be expressed as = - k k E k T exp[ / ] i i i B 0 , where E i and k i 0 are the reaction barrier and the pre-exponential factor, respectively. In our kMC simulations, all the transitions are assumed to have the same pre-exponential factor, i.e. uni2261 = - k k 10 s i 0 0 13 1 . 4.2.1. SIA diffusion trajectories. Starting from the origin at (0,0,0), the positions of the SIA are traced as a function of time. Figure  12 ( a ) shows two SIA diffusion trajectories up to 100 ps, at the temperatures of 300 K and 500 K. It can be seen from figure  12 ( a ) that the diffusion trajecto- ries consist of ‘nodes’, which are connected by line segments. From the projections on the basal plane (figure 12 ( b )), it is clear that the ‘nodes’ are in the form of hexagons, which rep- resent the direct O-O hopping mechanism on the basal plane. The short line segments connecting the neighboring hexagons are the O-M2-O migrations, while the long segments are the 1D motion via BC-BC glide mechanism. 4.2.2. Mean square displacement (MSD) of SIA diffusion. The mean square displacement (MSD) of the self-interstitial atoms is calculated similarly to the vacancy MSD described above. Notice that the interstitial atom being traced during the MSD simulations is not necessarily the same initial interstitial atom, because the O-M2-O mechanism is an interstitialcy mecha- nism and actually switches the initial interstitial atom with a lattice atom. In other words, overall, the MSD of the mass transport is being calculated here, from 100 K up to 900 K. The statistics have been demonstrated to be extremely important in calculating the diffusivities [ 56 ]. In this work, to increase the statistics, under each temperature 1000 kMC simulations were employed. In each kMC simulation, we collected the MSD data until 10 ns. Figures 13 ( a )–( c ) show the average MSD plots over 1000 kMC simulations, at 150 K, 600 K, and 900 K, respectively. It can be seen from the figure that the slope of <a> MSD is slightly steeper than that of <c> MSD. The ratio between the c -axis and a -axis diffusivities, Dc/Da, is above 0.8 for the Figure 11. ( a ) The total MSD for vacancy migration and its projection on the <a> and <c> directions in hcp Zr at 1000 K. The total MSD is defined as = + + MSD r t r t r t ( ) ( ) ( ) tot x y z 2 2 2 , where x - y plane represents the basal plane and z represents the <c> direction. The MSD in <a> direction is therefore calculated as r t ( ) x 2 , and the MSD in <c> direction is calculated as r t ( ) z 2 . The unit of MSD is the square of the lattice parameter on the basal plane, a 2 . It can be seen that uni2248 + < > < > MSD MSD MSD 2 tot a c . ( b ) The ratio between the diffusivities along the <c> and < a > directions (Dc/Da) at various temperatures. All the results in ( a ) and ( b ) are based on the assumption that the pre-factors in the hopping rate of the two paths are the same.
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  • Summer '19
  • Transition state, KMC

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